Unitary matrix explained

In linear algebra, an invertible complex square matrix is unitary if its matrix inverse equals its conjugate transpose, that is, if

$U^* U = UU^* = I,$

where is the identity matrix.

In physics, especially in quantum mechanics, the conjugate transpose is referred to as the Hermitian adjoint of a matrix and is denoted by a dagger (†), so the equation above is written

$U^\dagger U = UU^\dagger = I.$

A complex matrix is special unitary if it is unitary and its matrix determinant equals .

For real numbers, the analogue of a unitary matrix is an orthogonal matrix. Unitary matrices have significant importance in quantum mechanics because they preserve norms, and thus, probability amplitudes.

Properties

For any unitary matrix of finite size, the following hold:

Given two complex vectors and, multiplication by preserves their inner product; that is, .
is normal (

U^*U=UU^*

is diagonalizable; that is, is unitarily similar to a diagonal matrix, as a consequence of the spectral theorem. Thus, has a decomposition of the form

U=VDV^*,

where is unitary, and is diagonal and unitary.

\left|\det(U)\right|=1

. That is,

\det(U)

will be on the unit circle of the complex plane.

Its eigenspaces are orthogonal.
can be written as, where indicates the matrix exponential, is the imaginary unit, and is a Hermitian matrix.

For any nonnegative integer, the set of all unitary matrices with matrix multiplication forms a group, called the unitary group .

Every square matrix with unit Euclidean norm is the average of two unitary matrices.^[1]

Equivalent conditions

If U is a square, complex matrix, then the following conditions are equivalent:^[2]

is unitary.

U^*

is unitary.

is invertible with

U^-1=U^*

The columns of

form an orthonormal basis of

\Complexⁿ

with respect to the usual inner product. In other words,

U^*U=I

The rows of

form an orthonormal basis of

\Complexⁿ

with respect to the usual inner product. In other words,

UU^*=I

is an isometry with respect to the usual norm. That is,

\|Ux\|₂=\|x\|₂

for all

x\in\Complexⁿ

, where

\|x\|_2 = \sqrt

is a normal matrix (equivalently, there is an orthonormal basis formed by eigenvectors of

) with eigenvalues lying on the unit circle.

Elementary constructions

2 × 2 unitary matrix

One general expression of a unitary matrix is

$U = \begin a & b \\ -e^ b^* & e^ a^* \\\end,\qquad\left| a \right|^2 + \left| b \right|^2 = 1\,$

which depends on 4 real parameters (the phase of, the phase of, the relative magnitude between and, and the angle). The form is configured so the determinant of such a matrix is $\det(U) = e^ ~.$

The sub-group of those elements

with

\det(U)=1

is called the special unitary group SU(2).

Among several alternative forms, the matrix can be written in this form: $\ U = e^ \begin e^ \cos \theta & e^ \sin \theta \\ -e^ \sin \theta & e^ \cos \theta \\\end\,$

where

e^i\alpha\cos\theta=a

and

e^i\beta\sin\theta=b ,

above, and the angles

\varphi,\alpha,\beta,\theta

can take any values.

By introducing

\alpha=\psi+\delta

and

\beta=\psi-\delta ,

has the following factorization:

$U = e^ \begin e^ & 0 \\ 0 & e^\end\begin \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \\\end \begin e^ & 0 \\ 0 & e^\end ~.$

This expression highlights the relation between unitary matrices and orthogonal matrices of angle .

Another factorization is^[3]

$U = \begin \cos \rho & -\sin \rho \\ \sin \rho & \;\cos \rho \\\end \begin e^ & 0 \\ 0 & e^\end\begin \;\cos \sigma & \sin \sigma \\ -\sin \sigma & \cos \sigma \\\end ~.$

Many other factorizations of a unitary matrix in basic matrices are possible.^[4] ^[5] ^[6] ^[7] ^[8] ^[9]

External links

- - Web site: Show that the eigenvalues of a unitary matrix have modulus 1 . . March 28, 2016 .

Notes and References

Chi-Kwong. Li . Edward. Poon. 10.1080/03081080290025507. Additive decomposition of real matrices. 2002. Linear and Multilinear Algebra. 50. 4. 321–326. 120125694 .
Book: Horn . Roger A. . Johnson . Charles R. . Matrix Analysis . . 9781139020411 . 2013 . 10.1017/CBO9781139020411 .
Führ . Hartmut . Rzeszotnik . Ziemowit . 2018 . A note on factoring unitary matrices . Linear Algebra and Its Applications . 547 . 32–44 . 10.1016/j.laa.2018.02.017 . 125455174 . 0024-3795. free .
Book: Williams, Colin P. . 2011 . Quantum gates . Explorations in Quantum Computing . 82 . Williams . Colin P. . Texts in Computer Science . London, UK . Springer . en . 10.1007/978-1-84628-887-6_2 . 978-1-84628-887-6.
Book: Nielsen . M.A. . Michael Nielsen . Chuang . Isaac . Isaac Chuang . 2010 . Quantum Computation and Quantum Information . . 978-1-10700-217-3 . Cambridge, UK . 43641333 . 20.
Barenco . Adriano . Bennett . Charles H. . Cleve . Richard . DiVincenzo . David P. . Margolus . Norman . Shor . Peter . Sleator . Tycho . Smolin . John A. . Weinfurter . Harald . 6 . 1995-11-01 . dmy-all . Elementary gates for quantum computation . . American Physical Society (APS) . 52 . 5 . 1050-2947 . 10.1103/physreva.52.3457 . 3457–3467, esp.p. 3465 . 9912645 . quant-ph/9503016 . 8764584 .
Marvian . Iman . 2022-01-10 . dmy-all . Restrictions on realizable unitary operations imposed by symmetry and locality . Nature Physics . 18 . 3 . 283–289 . 2003.05524 . 10.1038/s41567-021-01464-0 . 245840243 . 1745-2481 . en .
Jarlskog . Cecilia . 2006 . Recursive parameterisation and invariant phases of unitary matrices . math-ph/0510034.
Alhambra, Álvaro M. . 10 January 2022 . Forbidden by symmetry . . 18 . 3 . 235–236 . 1745-2481 . 10.1038/s41567-021-01483-x . 256745894 . News & Views . The physics of large systems is often understood as the outcome of the local operations among its components. Now, it is shown that this picture may be incomplete in quantum systems whose interactions are constrained by symmetries..